We revisit, in an original and challenging perspective, the problem of testing the null hypothesis that the mode of a directional signal is equal to a given value. Motivated by a real data example where the signal is weak, we consider this problem under asymptotic scenarios for which the signal strength goes to zero at an arbitrary rate η n . Both under the null and the alternative, we focus on rotationally symmetric distributions. We show that, while they are asymptotically equivalent under

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... ed signal strength, the classical Wald and Watson tests exhibit very different (null and nonnull) behaviours when the signal becomes arbitrarily weak. To fully characterize how challenging the problem is as a function of η n , we adopt a Le Cam, convergenceof-statistical-experiments, point of view and show that the resulting limiting experiments crucially depend on η n . In the light of these results, the Watson test is shown to be adaptively rate-consistent and essentially adaptively Le Cam optimal. Throughout, our theoretical findings are illustrated via Monte-Carlo simulations. The practical relevance of our results is also shown on the real data example that motivated the present work.